A method for multi-objective optimization design of dual-plane coils with active learning assistance

Abstract

In view of the problems of strong multi-parameter coupling, time-consuming repeated calculation of physical model, low efficiency of multi-objective collaborative optimization and insufficient prediction accuracy of the optimal region of the proxy model in the existing dual-plane coil design method, the present application provides a dual-plane coil multi-objective optimization design method assisted by active learning, generates sample data by constructing a dual-plane coil physical model, uses an XGB model as a proxy model, and embeds it in NSGA-II for multi-objective optimization. At the same time, an active learning strategy is introduced, and the low uniformity error samples and high prediction uncertainty samples in the Pareto optimal solution set are returned to the physical model for recalculation, thereby continuously updating the training data and the proxy model. The method of the present application can reduce repeated high-precision physical calculation, improve the efficiency of multi-parameter collaborative optimization, and has higher design efficiency and optimization stability compared with conventional methods.

Description

Technical Field

[0001] This invention belongs to the field of magnetic compensation system design technology for atomic spin magnetometers, specifically relating to an active learning-assisted multi-objective optimization design method for dual-plane coils. Background Technology

[0002] Quantum precision measurement sensors such as atomic magnetometers typically require highly uniform magnetic field environments. As the core component of a magnetic field compensation system, the structure and parameter design of the magnetic field compensation coil directly affect the uniformity, stability, and measurement accuracy of the magnetic field within the target region. Dual-plane coils, with their open structure and ease of integration, are suitable for generating uniform magnetic fields in sensors such as atomic magnetometers. Currently, dual-plane coils are usually designed using the target field method, which involves first defining the desired magnetic field distribution within the target region, then inverting the current density on the coil plane to obtain the desired result, and finally discretizing the continuous current density into the actual winding path.

[0003] However, the design results of a dual-plane coil are affected by multiple parameters, such as the plane side length, plane spacing, regularization parameter, and current density expansion order. Furthermore, there are interdependent relationships between magnetic field uniformity, coil structural complexity, and current intensity cost. Traditional optimization methods typically require repeated calls to the physical model for magnetic field calculations during the solution process, resulting in high computational costs and making it difficult to efficiently achieve multi-parameter, multi-objective collaborative optimization.

[0004] Machine learning, as a surrogate model, can reduce repetitive physical computations to some extent. However, when relying solely on fixed training samples, the prediction accuracy and stability of the surrogate model near the optimal region may still be insufficient. Therefore, there is an urgent need for a dual-plane coil optimization design method that combines physical models, machine learning surrogate models, multi-objective optimization algorithms, and active learning strategies to improve optimization efficiency and design reliability. Summary of the Invention

[0005] To address the problems of strong multi-parameter coupling, time-consuming repetitive calculations of the physical model, low efficiency of multi-objective collaborative optimization, and insufficient prediction accuracy of the surrogate model in the optimal region in existing dual-plane coil design methods, this invention provides an active learning-assisted multi-objective optimization design method for dual-plane coils. Sample data is generated by constructing a dual-plane coil physical model, and the XGB model is used as a surrogate model, embedded into NSGA-II for multi-objective optimization. Simultaneously, an active learning strategy is introduced, using low-uniformity error samples and high prediction uncertainty samples from the Pareto optimal solution set to recalculate the physical model, thereby continuously updating the training data and the surrogate model, improving optimization efficiency, prediction accuracy, and search stability.

[0006] The technical solution of the present invention is as follows:

[0007] An active learning-assisted multi-objective optimization design method for dual-plane coils, characterized by the following steps:

[0008] Step 1: Determine the parameters to be optimized and the constraint range. The parameters to be optimized include: L, D, M, N, and λ. L is the half-side length of the dual-plane coil, D is the half-pitch of the dual-plane coil, M and N are both current density expansion orders, M and N are both positive integers in the preset integer set, and λ is a regularization parameter.

[0009] Step 2: Generate initial sample points within the constraints of the parameters to be optimized using Latin hypercube sampling;

[0010] Step 3: Calculate using the physical model of the two-plane coil. , ,and , It is an error in magnetic field uniformity. It is a high-frequency complexity. It comes at the cost of increased current intensity;

[0011] Step 4: Construct an initial sample set based on the dual-plane coil physical model;

[0012] Step 5: Train the XGB ensemble agent model using the initial sample set. The XGB ensemble agent model is an ensemble average model composed of multiple XGBs.

[0013] Step 6, use R 2 The accuracy of the XGB integrated proxy model is evaluated using RMSE. 2 It is the coefficient of determination, and RMSE is the root mean square error;

[0014] Step 7: Embed the XGB integrated agent model into NSGA-II to form a fast prediction model in the optimization process, so as to... , and Multi-objective optimization is performed using the optimization objective as the goal, and with Using a preset threshold as a feasible constraint, the current Pareto optimal solution set is obtained.

[0015] Step 8, Filtering Feasible solutions that are less than a preset threshold;

[0016] Step 9, Calculate ,and , It is the change in volume. It is the model's prediction of the amount of change;

[0017] Step 10, determine if If the value is less than the first convergence threshold, proceed to step 13 if no, and proceed to step 11 if yes.

[0018] Step 11, Determine Is it less than the second convergence threshold? If not, proceed to step 13; if yes, proceed to step 12.

[0019] Step 12: Determine whether the preset number of rounds is met consecutively. If not, proceed to step 13; if yes, proceed to step 14.

[0020] Step 13: Select candidate samples with low uniformity error and candidate samples with high prediction uncertainty, and then return to step 3 to enter the k+1 round loop, where k is a positive integer;

[0021] Step 14: After obtaining the final design parameters, calculate the flow function distribution on the plane of the dual-plane coil according to the current density expansion coefficient, extract the flow function contour lines according to the preset current interval, and convert the contour lines into discrete winding trajectories to obtain a manufacturable dual-plane coil structure.

[0022] Step 3 includes the following expression:

[0023] ,

[0024] ,

[0025] ,

[0026] ,

[0027] ,

[0028] ,

[0029] ,

[0030] in For any point within the target area The y-axis component of the magnetic induction intensity along the direction of the target magnetic field. Center point of the target area The y-axis component of the magnetic induction intensity along the direction of the target magnetic field. For intermediate values, m and n are both ordinal numbers. It is an intermediate quantity. The current density expansion factor is... For numerically stable terms, and All of these are wavenumbers corresponding to trigonometric basis functions.

[0031] Step 9 includes the following expression:

[0032] ,

[0033]

[0034] in, Let hypervolume be the Pareto optimal solution set obtained in the k-th round of active learning iteration. The hypervolume corresponding to the Pareto optimal solution set obtained in the (k-1)th round of active learning iteration; To optimize the number of objectives, t is the index number. Let i be the number of candidate solutions in the current Pareto optimal solution set, where i is the index. For the k-th round XGB integrated proxy model, the i-th candidate solution The predicted value for the t-th optimization objective. Let be the predicted value of the i-th candidate solution for the t-th optimization objective by the XGB ensemble surrogate model in the (k-1)-th round. and These are the maximum and minimum values ​​of the t-th optimization objective in the current Pareto optimal solution set, respectively.

[0035] include:

[0036] .

[0037] include:

[0038] .

[0039] The two coil planes in the dual-plane coil are arranged symmetrically with respect to the center of the target region, and the target region is located between the two coil planes. When the dual-plane coil generates a uniform magnetic field along the y-axis, the current directions on the two coil planes are opposite.

[0040] Step 13 includes: determining the prediction uncertainty based on the prediction dispersion of multiple XGB sub-models for the same candidate sample; for the same candidate sample, calculating the prediction standard deviation of multiple XGB sub-models on each optimization objective, and averaging the prediction standard deviations after normalization to obtain the comprehensive prediction uncertainty of the candidate sample.

[0041] Step 13 includes: for cases where the convergence condition is not met, selecting the top solutions from the current Pareto optimal set, sorted by magnetic field uniformity error from smallest to largest. 10 candidate samples It is a positive integer, and the first [number] are sorted by prediction uncertainty from largest to smallest. 10 candidate samples The value is a positive integer, and duplicate candidate samples are deduplicated to obtain samples to be updated. The samples to be updated are input into the dual-plane coil physical model for recalculation to obtain the corresponding real performance index. The samples to be updated and their real performance index are added to the sample set as new samples to update the sample set and retrain the XGB ensemble proxy model.

[0042] A uniform field dual-plane coil structure is characterized by comprising two coil planes arranged opposite to each other and parallel to each other, with a target region formed between the two coil planes; each coil plane includes multiple conductor trajectories formed by discretization of the stream function contour lines, wherein the plane side length, plane spacing, regularization parameter and current density expansion order corresponding to the conductor trajectories are determined by the above-mentioned active learning-assisted dual-plane coil multi-objective optimization design method.

[0043] The technical effects of this invention are as follows: This invention provides an active learning-assisted multi-objective optimization design method for dual-plane coils. It mainly revolves around the multi-objective optimization design of dual-plane coils. The core idea is to combine the physical model of dual-plane coils, the XGB integrated surrogate model, the NSGA-II multi-objective optimization algorithm, and the active learning strategy. By selecting candidate samples with low magnetic field uniformity error and candidate samples with high prediction uncertainty to recalculate the physical model, the efficiency and stability of multi-parameter collaborative optimization of dual-plane coils are improved.

[0044] The advantages of this invention compared to existing technologies are:

[0045] (1) The present invention uses the XGB proxy model to replace a large number of repetitive physical model calculations in the optimization process, which can significantly reduce the calculation cost of multi-parameter optimization of dual-plane coils and improve design efficiency.

[0046] (2) The present invention embeds the XGB proxy model into NSGA-II, and at the same time takes magnetic field uniformity error, high frequency complexity and current intensity cost as optimization targets, which can achieve synergistic optimization between magnetic field performance, coil structure complexity and current cost.

[0047] (3) The present invention introduces an active learning strategy, which returns the low uniformity error samples and high prediction uncertainty samples in the Pareto optimal solution set to the physical model for recalculation and updates the training dataset, so that the surrogate model can continuously improve the prediction accuracy in the key design region. Attached Figure Description

[0048] Figure 1 This is a schematic diagram of the process of implementing the active learning-assisted dual-plane coil multi-objective optimization design method of the present invention. Figure 1The algorithm includes a start section, a dataset section, an XGB-NSGA-II section (XGB stands for Extreme Gradient Boosting Model, and NSGA-II stands for Second Generation Non-Dominated Sorting Genetic Algorithm or Non-Dominated Sorting Genetic Algorithm II), an active learning section, and a conclusion. The dataset section includes step 1, determining the parameters to be optimized and the constraints (including L, D, M, N, λ, where L is the half-side length of the dual-plane coil, D is the half-spacing of the dual-plane coil, M and N are the current density expansion orders, M and N are positive integers from a preset set of integers, and λ is the regularization parameter); step 2, generating initial sample points based on Latin hypercube sampling; step 3, calculating the dual-plane coil physical model; and step 4, constructing the initial sample set. The XGB-NSGA-II section includes step 5, training the XGB ensemble surrogate model (the XGB ensemble surrogate model is an ensemble average model composed of multiple XGBs); and step 6, using R... 2 The accuracy of the XGB integrated proxy model was evaluated using RMSE. 2 The coefficient of determination (RMSE is the root mean square error); Step 7, use the XGB integrated surrogate model to accelerate NSGA-II (that is, embed the XGB integrated surrogate model into NSGA-II as a fast prediction model in the optimization process); Step 8, screen feasible solutions with magnetic field uniformity error less than a preset threshold; Step 9, calculate the hypervolume change and the model predicted change. The active learning part includes Step 10, judging whether the hypervolume change is less than the first convergence threshold. If not, proceed to Step 13; if yes, proceed to Step 11; Step 11, judging whether the model predicted change is less than the second convergence threshold. If not, proceed to Step 13; if yes, proceed to Step 12; Step 12, judging whether the preset number of rounds is continuously satisfied. If not, proceed to Step 13; if yes, proceed to Step 14; Step 13, select candidate samples with low uniformity error and candidate samples with high prediction uncertainty, and then return to Step 3 to enter k+1 rounds of loop, where k is a positive integer; Step 14, output the Pareto front and the final design point of the dual-plane coil.

[0049] Figure 2 This is a schematic diagram of the discrete winding structure of a dual-plane coil obtained by the active learning-assisted dual-plane coil multi-objective optimization design method of the present invention. Figure 2 In the diagram, 1 represents the target region, 2 represents the target field selection region, 3 represents the conductor trajectory obtained by discretizing the stream function contour lines, xyz represents the three-dimensional rectangular coordinate axes (i.e., x-axis, y-axis, and z-axis), a represents half the side length of the target region (cube-shaped) (i.e., 2a is the side length of the target region), D represents the half-spacing of the dual-plane coils (i.e., 2D is the spacing between the left and right plane coils), and L represents half the side length of the dual-plane coils (both the left and right plane coils are squares, and 2L is the side length of the dual-plane coils). Detailed Implementation

[0050] The following is in conjunction with the attached diagram ( Figures 1-2 The present invention will be described in conjunction with the examples.

[0051] Figure 1 This is a schematic diagram of the process of implementing the active learning-assisted dual-plane coil multi-objective optimization design method of the present invention. Figure 2 This is a schematic diagram of the discrete winding structure of a dual-plane coil obtained by the active learning-assisted multi-objective optimization design method for dual-plane coils according to the present invention. (Reference) Figures 1-2 As shown, an active learning-assisted multi-objective optimization design method for dual-plane coils includes the following steps: Step 1, determining the parameters to be optimized and their constraint range, wherein the parameters to be optimized include: L, D, M, N, and λ, where L is the half-side length of the dual-plane coil, D is the half-pitch of the dual-plane coil, M and N are both current density expansion orders, M and N are both positive integers in a preset set of integers, and λ is a regularization parameter; Step 2, generating initial sample points within the constraint range of the parameters to be optimized using Latin hypercube sampling;

[0052] Step 3: Calculate using the physical model of the two-plane coil. , ,and , It is an error in magnetic field uniformity. It is a high-frequency complexity. It comes at the cost of increased current intensity;

[0053] Step 4: Construct an initial sample set based on the dual-plane coil physical model; Step 5: Train the XGB ensemble surrogate model using the initial sample set. The XGB ensemble surrogate model is an ensemble average model composed of multiple XGBs; Step 6: Use R... 2 The accuracy of the XGB integrated proxy model is evaluated using RMSE. 2 It is the coefficient of determination, and RMSE is the root mean square error;

[0054] Step 7: Embed the XGB integrated agent model into NSGA-II to form a fast prediction model in the optimization process, so as to... , and Multi-objective optimization is performed using the optimization objective as the goal, and with Using a preset threshold as a feasible constraint, the current Pareto optimal solution set is obtained.

[0055] Step 8, Filtering Feasible solutions less than the preset threshold; Step 9, calculate ,and , It is the change in volume. It is the model's prediction of the amount of change;

[0056] Step 10, determine if If the value is less than the first convergence threshold, proceed to step 13 if not, and proceed to step 11 if yes; Step 11, determine... Is it less than the second convergence threshold? If not, proceed to step 13; if yes, proceed to step 12. Step 12: Determine if the preset number of rounds is met continuously. If not, proceed to step 13; if yes, proceed to step 14.

[0057] Step 13: Select candidate samples with low uniformity error and candidate samples with high prediction uncertainty, and then return to step 3 to enter the k+1 round loop, where k is a positive integer;

[0058] Step 14: After obtaining the final design parameters, calculate the flow function distribution on the plane of the dual-plane coil according to the current density expansion coefficient, extract the flow function contour lines according to the preset current interval, and convert the contour lines into discrete winding trajectories to obtain a manufacturable dual-plane coil structure.

[0059] Step 3 includes the following expression:

[0060] ,

[0061] ,

[0062] ,

[0063] ,

[0064] ,

[0065] ,

[0066] ,

[0067] in For any point within the target area The y-axis component of the magnetic induction intensity along the direction of the target magnetic field. Center point of the target area The y-axis component of the magnetic induction intensity along the direction of the target magnetic field. For intermediate values, m and n are both ordinal numbers. It is an intermediate quantity. The current density expansion factor is... For numerically stable terms, and All of these are wavenumbers corresponding to trigonometric basis functions.

[0068] Step 9 includes the following expression:

[0069] ,

[0070]

[0071] in, Let hypervolume be the Pareto optimal solution set obtained in the k-th round of active learning iteration. The hypervolume corresponding to the Pareto optimal solution set obtained in the (k-1)th round of active learning iteration; To optimize the number of objectives, t is the index number. Let i be the number of candidate solutions in the current Pareto optimal solution set, where i is the index. For the k-th round XGB integrated proxy model, the i-th candidate solution The predicted value for the t-th optimization objective. Let be the predicted value of the i-th candidate solution for the t-th optimization objective by the XGB ensemble surrogate model in the (k-1)-th round. and These are the maximum and minimum values ​​of the t-th optimization objective in the current Pareto optimal solution set, respectively.

[0072] include:

[0073] .

[0074] include:

[0075] .

[0076] The two coil planes in the dual-plane coil are arranged symmetrically with respect to the center of the target region, and the target region is located between the two coil planes. When the dual-plane coil generates a uniform magnetic field along the y-axis, the current directions on the two coil planes are opposite.

[0077] Step 13 includes: determining the prediction uncertainty based on the prediction dispersion of multiple XGB sub-models for the same candidate sample; for the same candidate sample, calculating the prediction standard deviation of multiple XGB sub-models on each optimization objective, and averaging the prediction standard deviations after normalization to obtain the comprehensive prediction uncertainty of the candidate sample.

[0078] Step 13 includes: for cases where the convergence condition is not met, selecting the top solutions from the current Pareto optimal set, sorted by magnetic field uniformity error from smallest to largest. 10 candidate samples It is a positive integer, and the first [number] are sorted by prediction uncertainty from largest to smallest. 10 candidate samples The value is a positive integer, and duplicate candidate samples are deduplicated to obtain samples to be updated. The samples to be updated are input into the dual-plane coil physical model for recalculation to obtain the corresponding real performance index. The samples to be updated and their real performance index are added to the sample set as new samples to update the sample set and retrain the XGB ensemble proxy model.

[0079] A uniform field dual-plane coil structure includes two oppositely arranged and parallel coil planes (which form conductor trajectories 3 obtained by discretization of stream function contour lines), and a target region 1 is formed between the two coil planes (the target region has a side length of 2a, and there is a smaller target field point selection region 2 within the target region); each coil plane includes multiple conductor trajectories formed by discretization of stream function contour lines, and the plane side length 2L, plane spacing 2D, regularization parameter, and current density expansion order (M and N) corresponding to the conductor trajectories are determined by the above-mentioned active learning-assisted dual-plane coil multi-objective optimization design method.

[0080] This invention relates to an active learning-assisted multi-objective optimization design method for biplane coils. Conventional biplane coil design methods typically rely on target field methods and traditional optimization algorithms, repeatedly calling physical models to calculate magnetic field distributions under different structural parameters to obtain coil parameters. However, such methods suffer from high computational cost and low optimization efficiency in multi-parameter, multi-objective collaborative design. The method of this invention first determines the parameters to be optimized and the constraint range of the biplane coil, generating an initial sample set based on the biplane coil physical model. Then, an ensemble surrogate model composed of multiple extreme gradient boosting (XGB) models is trained as a surrogate model, and this surrogate model is embedded into the non-dominated sorting genetic algorithm II (NSGA-II). Furthermore, an active learning strategy is used to update the sample set and the surrogate model, improving the prediction accuracy of the XGB model in key design regions and the search stability of NSGA-II. The method of this invention reduces repetitive high-precision physical calculations, improves the efficiency of multi-parameter collaborative optimization, and exhibits higher design efficiency and optimization stability compared to conventional methods.

[0081] An active learning-assisted multi-objective optimization design method for biplane coils includes the following steps:

[0082] Step 1: Determine the parameters to be optimized and the constraint range of the dual-plane coil.

[0083] Step 2: Generate an initial sample set based on the dual-plane coil physical model. Latin hypercube sampling is used to generate sample points within the constraints of the parameters to be optimized. The physical model is then used to calculate the magnetic field uniformity error corresponding to each sample point. High frequency complexity and current intensity cost ;

[0084] Step 3: Train an XGB proxy model based on the initial sample set. The proxy model is an ensemble average model composed of multiple XGB models.

[0085] Step 4: Embed the XGB integrated surrogate model into NSGA-II to address magnetic field uniformity error. High frequency complexity and current intensity cost Multi-objective optimization is performed with the magnetic field uniformity error being less than a preset uniformity error threshold as a feasible constraint to obtain the current Pareto optimal solution set.

[0086] Step 5: Calculate the hypervolume change in adjacent iterations. and model-predicted change and based on Less than the first convergence threshold, Convergence is determined by the number of rounds in which the convergence condition is met consecutively and the second convergence threshold is less than the threshold.

[0087] Step 6: If the convergence condition is not met, select the top solutions from the current Pareto optimal set, sorted by magnetic field uniformity error from smallest to largest. The candidate samples, and the top ones sorted by prediction uncertainty from largest to smallest. The XGB ensemble proxy model is retrained by taking 10 candidate samples and removing duplicates. The sample to be updated is then input into the dual-plane coil physical model for recalculation to obtain the corresponding real performance index. The sample to be updated and its real performance index are then added to the sample set as new samples to update the sample set and retrain the XGB ensemble proxy model.

[0088] Step 7: Repeat steps 4 to 6 until the convergence condition is met;

[0089] Step 8: After obtaining the final design parameters, calculate the flow function distribution on the plane of the dual-plane coil according to the current density expansion coefficient, extract the flow function contour lines according to the preset current interval, and convert the contour lines into discrete winding trajectories to obtain a manufacturable dual-plane coil structure.

[0090] The constraint range of the parameters to be optimized includes: the side length of the biplane coil. Within a preset side length range, the spacing between the dual-plane coils Within the preset spacing range, the regularization parameter Within the preset regularization range, the order of the current density expansion and It is a positive integer in the preset set of integers, and .

[0091] In step 3, XGB is an extreme gradient boosting model used to train and obtain the parameters to be optimized for the dual-plane coil and the magnetic field uniformity error based on sample data. High frequency complexity and current intensity cost The XGB surrogate model is used for training and validation using training and test sets. The prediction performance is evaluated using the coefficient of determination (R²) and root mean square error (RMSE). The XGB surrogate model is an ensemble average model consisting of 10 XGB models. The NSGA-II is a non-dominated sorting genetic algorithm II used for multi-objective optimization of the biplane coil structure parameters based on the surrogate model, obtaining a Pareto optimal solution set.

[0092] In step 3, to obtain the prediction uncertainty of the candidate samples, this embodiment uses multiple XGB sub-models to form an integrated surrogate model. For the same candidate sample, each XGB sub-model outputs the corresponding magnetic field uniformity error. High frequency complexity and current intensity cost The predicted values ​​of the indicators. If the prediction results of multiple XGB sub-models for the same candidate sample differ significantly, it indicates that the candidate sample is in a region of high prediction uncertainty of the surrogate model; if the prediction results differ slightly, it indicates that the surrogate model has high prediction stability near the candidate sample. Therefore, this invention uses the standard deviation of the prediction results of multiple XGB sub-models to characterize the prediction uncertainty of the candidate sample, and normalizes the results for different optimization objectives to obtain the comprehensive prediction uncertainty.

[0093] The magnetic field uniformity error High frequency complexity and current intensity cost They are represented as follows:

[0094] ,

[0095] ,

[0096] ,

[0097] ,

[0098] in, Let be the magnetic flux density component along the direction of the target magnetic field at any point within the target area. This represents the magnetic flux density component along the direction of the target magnetic field at the center point of the target region. , , and These are the structural parameters involved in the magnetic field distribution calculation in the physical model of the two-plane coil. Here are the Tikhonov regularization parameters. Calculate... , , . and These are the wave numbers corresponding to the trigonometric basis functions. This is an intermediate quantity in the formula. The current density expansion factor is... For numerically stable terms, It is the half-side length of the dual-plane coil.

[0099] The XGB is an extreme gradient boosting model used to train and obtain the parameters to be optimized for the dual-plane coil and the magnetic field uniformity error based on sample data. High frequency complexity and current intensity cost The surrogate model between them; the NSGA-II is a non-dominated sorting genetic algorithm II, used to perform multi-objective optimization of the biplane coil structure parameters based on the surrogate model and obtain the Pareto optimal solution set.

[0100] The XGB is trained and validated using training and testing sets, and employs the coefficient of determination. and root mean square error The prediction performance is evaluated, and the XGB proxy model is an ensemble average model composed of multiple XGB models.

[0101] The prediction uncertainty is determined based on the degree of dispersion of predictions for the same candidate sample by multiple XGB sub-models. For the same candidate sample, the prediction standard deviations of multiple XGB sub-models on each optimization objective are calculated, and the prediction standard deviations are normalized and averaged to obtain the comprehensive prediction uncertainty of the candidate sample.

[0102] The hypervolume change and model-predicted change Represented as:

[0103] ,

[0104] ,

[0105] in, Let hypervolume be the Pareto optimal solution set obtained in the k-th round of active learning iteration. The hypervolume corresponding to the Pareto optimal solution set obtained in the (k-1)th round of active learning iteration. This represents the change in hypervolume between two adjacent rounds of active learning iterations. This represents the number of candidate solutions in the current Pareto optimal solution set. Let be the predicted value of the i-th candidate solution by the XGB surrogate model in the k-th round, for the t-th optimization objective. Let be the predicted value of the i-th candidate solution for the t-th optimization objective by the XGB surrogate model in the (k-1)-th round. and These are the maximum and minimum values ​​of the t-th optimization objective in the current Pareto optimal solution set, respectively. It is an active learning iteration cycle. It is the i-th candidate solution. To optimize the number of targets, in this method, .

[0106] In step 6, candidate samples with smaller magnetic field uniformity errors are closer to the target design region. Returning these samples to the dual-plane coil physical model for recalculation helps improve the local prediction accuracy of the surrogate model in the high-performance design region. Candidate samples with higher prediction uncertainty correspond to regions where the surrogate model lacks sufficient understanding. Returning these samples to the physical model for recalculation helps reduce the prediction bias of the surrogate model near the Pareto front. By simultaneously selecting both types of samples, this invention can balance local development capabilities and global exploration capabilities, thereby improving the stability and reliability of the dual-plane coil multi-objective optimization process.

[0107] The dual-plane coil includes two parallel and symmetrically arranged coil planes, which are symmetrically arranged with respect to the center of the target region, which is located between the two coil planes; when generating a uniform magnetic field along the y-axis, the current directions on the two coil planes are opposite.

[0108] A uniform field dual-plane coil structure includes two coil planes arranged opposite to each other and parallel to each other, forming a target region between the two coil planes; each coil plane includes multiple conductor trajectories formed by discretization of the stream function contour lines, and the plane side length, plane spacing, regularization parameter and current density expansion order corresponding to the conductor trajectories are determined by the above method.

[0109] like Figure 1 As shown, this invention provides an active learning-assisted multi-objective optimization design method for dual-plane coils. This method is based on a dual-plane coil physical model, utilizes Latin hypercube sampling to establish an initial sample set, employs 10 XGB models to construct an integrated surrogate model, and embeds this integrated surrogate model into NSGA-II to address magnetic field uniformity errors. High frequency complexity and current intensity cost The optimization objective is used as the target for optimization. In the optimization process, an active learning strategy is further introduced. Candidate samples with small magnetic field uniformity errors and high prediction uncertainties in the current Pareto optimal solution set are returned to the physical model for recalculation, thereby updating the sample set and retraining the surrogate model until the convergence condition is met, and finally outputting the design parameters of the dual-plane coil.

[0110] An active learning-assisted multi-objective optimization design method for biplane coils includes the following steps:

[0111] Step 1: Determine the parameters to be optimized and the constraint range of the dual-plane coil;

[0112] A dual-plane coil structure model is established. The dual-plane coil comprises two parallel and symmetrically arranged coil planes, which are symmetrically positioned with respect to the center of the target region. The target region lies between the two coil planes. The side lengths of the two coil planes are... The distance between the two coil planes is .

[0113] A Cartesian coordinate system is established with the center of the target region as the origin. When a dual-plane coil is used to generate a uniform magnetic field along the y-axis, the current directions on the two coil planes are opposite; when a dual-plane coil is used to generate a uniform magnetic field along the x-axis, the current directions on the two coil planes are the same. This embodiment uses a dual-plane coil generating a uniform magnetic field along the y-axis as an example. The parameters to be optimized include the side lengths of the planes of the dual-plane coil. Biplane spacing Regularization parameters and the order of current density expansion and To balance computational accuracy and coil structural complexity, it is preferable to let... .

[0114] In one specific implementation, the range of the parameter to be optimized is: , , and ,and The above parameter ranges can be adjusted according to the target area size, coil processing size, target magnetic field direction, and magnetic field uniformity requirements.

[0115] Step 2: Generate an initial sample set based on the dual-plane coil physical model;

[0116] The discrete conductor distribution on the plane of the two-plane coil is approximated as a continuous current sheet, and the current density vector is decomposed into y-axis and z-axis components. Using trigonometric functions as basis functions, a series expansion of the current density components is performed to establish the physical model of the two-plane coil. The current density on the plane of the two-plane coil is expanded as follows:

[0117] ,

[0118] ,

[0119] ,

[0120] ,

[0121] in, Let be the component of the current density in the y-axis direction. Let be the component of the current density along the z-axis. The current density expansion factor is... and Let be the order of the current density expansion. and These represent the wavenumbers corresponding to the trigonometric basis functions. Based on the target field method, a target magnetic field distribution is pre-defined within the target region, and the current density expansion coefficients on the coil plane are solved using a two-plane coil physical model. To improve the solution stability, the Tikhonov regularization method is used to solve for the current density expansion coefficients. Its solution form is:

[0122] ,

[0123] in, The coefficient vector is composed of the current density expansion coefficients. The coefficient matrix is ​​determined by the locations of magnetic field points within the target region and the physical model of the two-plane coils. A vector composed of preset magnetic field values ​​within the target area. For the parameters of the Tikhonov regularization method, This is the normalized penalty matrix.

[0124] Within the constraints of the parameters to be optimized, initial sample points are generated using Latin hypercube sampling. For each set of sample points, the magnetic field distribution within the target region is calculated using a dual-plane coil physical model, and the corresponding output parameters are further calculated. The output parameters include the magnetic field uniformity error. High frequency complexity and current intensity cost The magnetic field uniformity error ε is expressed as:

[0125] ,

[0126] ,

[0127] ,

[0128] ,

[0129] in, Let be the magnetic flux density component along the direction of the target magnetic field at any point within the target area. This represents the magnetic flux density component along the direction of the target magnetic field at the center point of the target region. , , and These are the structural parameters involved in the magnetic field distribution calculation in the physical model of the two-plane coil. Here are the Tikhonov regularization parameters. Calculate... , , . and These are the wave numbers corresponding to the trigonometric basis functions. This is an intermediate quantity in the formula. The current density expansion factor is... For numerically stable terms, This represents the half-side length of the dual-plane coil. These three output parameters allow for the simultaneous evaluation of the magnetic field uniformity error of the dual-plane coil. High frequency complexity and current intensity cost This provides an objective function for subsequent multi-objective optimization. In one specific implementation, the initial sample set includes 1200 sets of sample data.

[0130] Step 3: Train the XGB ensemble agent model;

[0131] Based on the initial sample set obtained in step 2, the XGB surrogate model is trained. The XGB surrogate model is used to learn the nonlinear mapping relationship between the parameters to be optimized and the output parameters, that is, to establish a path from input parameters to output parameters. The mapping relationship between them.

[0132] To improve the prediction stability and generalization ability of the surrogate model, this embodiment uses 10 XGB models to form an ensemble averaging model. Specifically, 10 XGB models are trained separately. For the same set of input parameters, the 10 XGB models output prediction results respectively. The 10 prediction results are then averaged to obtain the final surrogate model output.

[0133] In one specific implementation, the sample set is divided into a training set and a test set in an 8:2 ratio. The coefficient of determination is used. and root mean square error Evaluate the predictive performance of the surrogate model. Used to characterize the degree of fit between the model's predictions and the actual calculations of the physical model. Used to characterize the average error between the predicted result and the actual result.

[0134] Step 4: Embed the XGB integrated proxy model into NSGA-II;

[0135] The XGB integrated surrogate model obtained in step 3 is embedded into NSGA-II as a fast prediction model during the optimization process. The magnetic field uniformity error is used as a metric. High frequency complexity and current intensity cost For multi-objective optimization, the feasible search region must satisfy the magnetic field uniformity error. If the value is less than 0.01, a Pareto optimal solution set is obtained;

[0136] During the optimization process, the NSGA-II algorithm searches for the Pareto optimal solution set through non-dominated sorting, elite preservation, and crowding distance mechanisms. The XGB model is embedded into NSGA-II to account for magnetic field homogeneity errors. High frequency complexity and current intensity cost For multi-objective optimization, the feasible search region must satisfy the magnetic field uniformity error. A value less than 0.01 yields the Pareto optimal solution set. Since the XGB ensemble surrogate model can quickly predict the output target corresponding to the candidate parameters, it is not necessary to call the high-precision physical model in each optimization iteration, thus significantly reducing the computational cost.

[0137] In one specific implementation, the NSGA-II population size is 400, and the maximum number of iterations is 200. Furthermore, to improve optimization efficiency, candidate solutions that satisfy the prediction magnetic field uniformity error of less than 0.01 are included as feasible candidate solutions in the feasible search region.

[0138] Step 5: Calculate the hypervolume change in adjacent iterations. and model-predicted change and based on and Determine whether the convergence occurs after three consecutive rounds of satisfying the condition.

[0139] Supervolume change and model-predicted change Represented as:

[0140]

[0141]

[0142] in, Let hypervolume be the Pareto optimal solution set obtained in the k-th round of active learning iteration. The hypervolume corresponding to the Pareto optimal solution set obtained in the (k-1)th round of active learning iteration. This represents the change in hypervolume between two adjacent rounds of active learning iterations. This represents the number of candidate solutions in the current Pareto optimal solution set. Let be the predicted value of the i-th candidate solution by the XGB surrogate model in the k-th round, for the t-th optimization objective. Let be the predicted value of the i-th candidate solution for the t-th optimization objective by the XGB surrogate model in the (k-1)-th round. and These are the maximum and minimum values ​​of the t-th optimization objective in the current Pareto optimal solution set, respectively. It is an active learning iteration cycle. It is the i-th candidate solution. To optimize the number of targets, in this method, .

[0143] Step 6: If the convergence condition is not met, select the top solutions from the current Pareto optimal set, sorted by magnetic field uniformity error from smallest to largest. The candidate samples, and the top ones sorted by prediction uncertainty from largest to smallest. A number of candidate samples are selected, and duplicate candidate samples are deduplicated to obtain samples to be updated. The samples to be updated are input into the dual-plane coil physical model for recalculation to obtain the corresponding real performance indicators, and are added to the sample set as new samples to update the sample set and retrain the surrogate model.

[0144] Step 7: Repeat steps 4 to 6 until the convergence condition is met;

[0145] Step 8: Output the Pareto optimal solution set and the final design parameters of the dual-plane coil, calculate the flow function distribution based on the final design parameters, extract the flow function contour lines according to the preset current interval, and convert the contour lines into discrete winding trajectories to obtain a manufacturable dual-plane coil structure.

[0146] Based on the design method described in this invention, a biplane coil with a relatively simple structure and small magnetic field uniformity error is designed, and its performance is then simulated using numerical simulation. The side length of the biplane coil is... 3.84cm, dual-plane spacing The current density expansion order is 1.59 cm. and All are 8. The Y-axis direction is the positive direction of the target magnetic field. The target region is a cube formed by dashed lines. The target field point is a cube enclosed by the positive semi-axes of the three coordinate axes and the target region, which occupies only 1 / 8 of the entire target region. Based on these parameters, discretization is performed using a stream function to obtain the structure of the dual-plane coil, as shown below. Figure 2 As shown. Figure 2 In the diagram, 1 represents the target region, 2 represents the target field selection region, and 3 represents the traverse trajectory obtained by discretizing the stream function contour lines.

[0147] Contents not described in detail in this specification are prior art known to those skilled in the art. It is hereby indicated that the above description is intended to help those skilled in the art understand this invention, but does not limit the scope of protection of this invention. Any equivalent substitutions, modifications, improvements, and / or simplifications of the above descriptions that do not depart from the essential content of this invention fall within the scope of protection of this invention.

Claims

1. A multi-objective optimization design method for dual-plane coils with active learning assistance, characterized in that, Includes the following steps: Step 1: Determine the parameters to be optimized and the constraint range. The parameters to be optimized include: L, D, M, N, and λ. L is the half-side length of the dual-plane coil, D is the half-pitch of the dual-plane coil, M and N are both current density expansion orders, M and N are both positive integers in the preset integer set, and λ is a regularization parameter. Step 2: Generate initial sample points within the constraints of the parameters to be optimized using Latin hypercube sampling; Step 3: Calculate using the physical model of the two-plane coil. , ,and , It is an error in magnetic field uniformity. It is a high-frequency complexity. It comes at the cost of increased current intensity; Step 4: Construct an initial sample set based on the dual-plane coil physical model; Step 5: Train the XGB ensemble agent model using the initial sample set. The XGB ensemble agent model is an ensemble average model composed of multiple XGBs. Step 6, use R 2 The accuracy of the XGB integrated proxy model is evaluated using RMSE. 2 It is the coefficient of determination, and RMSE is the root mean square error; Step 7: Embed the XGB integrated agent model into NSGA-II to form a fast prediction model in the optimization process, so as to... , and Multi-objective optimization is performed using the optimization objective as the goal, and with Using a preset threshold as a feasible constraint, the current Pareto optimal solution set is obtained. Step 8, Filtering Feasible solutions that are less than a preset threshold; Step 9, Calculate ,and , It is the change in volume. It is the model's prediction of the amount of change; Step 10, determine if If the value is less than the first convergence threshold, proceed to step 13 if no, and proceed to step 11 if yes. Step 11, Determine Is it less than the second convergence threshold? If not, proceed to step 13; if yes, proceed to step 12. Step 12: Determine whether the preset number of rounds is met consecutively. If not, proceed to step 13; if yes, proceed to step 14. Step 13: Select candidate samples with low uniformity error and candidate samples with high prediction uncertainty, and then return to step 3 to enter the k+1 round loop, where k is a positive integer; Step 14: After obtaining the final design parameters, calculate the flow function distribution on the plane of the dual-plane coil according to the current density expansion coefficient, extract the flow function contour lines according to the preset current interval, and convert the contour lines into discrete winding trajectories to obtain a manufacturable dual-plane coil structure.

2. The active learning-assisted multi-objective optimization design method for dual-plane coils according to claim 1, characterized in that, Step 3 includes the following expression: ， ， ， ， ， ， ， in For any point within the target area The y-axis component of the magnetic induction intensity along the direction of the target magnetic field. Center point of the target area The y-axis component of the magnetic induction intensity along the direction of the target magnetic field. For intermediate values, m and n are both ordinal numbers. It is an intermediate quantity. The current density expansion factor is... For numerically stable terms, and All of these are wavenumbers corresponding to trigonometric basis functions.

3. The active learning-assisted multi-objective optimization design method for dual-plane coils according to claim 1, characterized in that, Step 9 includes the following expression: ， 4. Among them, Let hypervolume be the Pareto optimal solution set obtained in the k-th round of active learning iteration. The hypervolume corresponding to the Pareto optimal solution set obtained in the (k-1)th round of active learning iteration; To optimize the number of objectives, t is the index number. Let i be the number of candidate solutions in the current Pareto optimal solution set, where i is the index. For the k-th round XGB integrated proxy model, the i-th candidate solution The predicted value for the t-th optimization objective. Let be the predicted value of the i-th candidate solution for the t-th optimization objective by the XGB ensemble surrogate model in the (k-1)-th round. and These are the maximum and minimum values ​​of the t-th optimization objective in the current Pareto optimal solution set, respectively.

5. The active learning-assisted multi-objective optimization design method for dual-plane coils according to claim 2, characterized in that, include: 。 6. The active learning-assisted multi-objective optimization design method for dual-plane coils according to claim 3, characterized in that, include: 。 7. The active learning-assisted multi-objective optimization design method for dual-plane coils according to claim 1, characterized in that, The two coil planes in the dual-plane coil are arranged symmetrically with respect to the center of the target region, and the target region is located between the two coil planes. When the dual-plane coil generates a uniform magnetic field along the y-axis, the current directions on the two coil planes are opposite.

8. The active learning-assisted multi-objective optimization design method for dual-plane coils according to claim 1, characterized in that, Step 13 includes: determining the prediction uncertainty based on the prediction dispersion of multiple XGB sub-models for the same candidate sample; for the same candidate sample, calculating the prediction standard deviation of multiple XGB sub-models on each optimization objective, and averaging the prediction standard deviations after normalization to obtain the comprehensive prediction uncertainty of the candidate sample.

9. The active learning-assisted multi-objective optimization design method for dual-plane coils according to claim 1, characterized in that, Step 13 includes: for cases where the convergence condition is not met, selecting the top solutions from the current Pareto optimal set, sorted by magnetic field uniformity error from smallest to largest. 10 candidate samples It is a positive integer, and the first [number] are sorted by prediction uncertainty from largest to smallest. 10 candidate samples The value is a positive integer, and duplicate candidate samples are deduplicated to obtain samples to be updated. The samples to be updated are input into the dual-plane coil physical model for recalculation to obtain the corresponding real performance index. The samples to be updated and their real performance index are added to the sample set as new samples to update the sample set and retrain the XGB ensemble proxy model.

10. A uniform field dual-plane coil structure, characterized in that, It includes two coil planes that are set opposite to each other and parallel to each other, and a target region is formed between the two coil planes; each coil plane includes multiple conductor trajectories formed by discretization of the stream function contour lines, and the plane side length, plane spacing, regularization parameter and current density expansion order corresponding to the conductor trajectories are determined by the active learning-assisted dual-plane coil multi-objective optimization design method of any one of claims 1-8.

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